32 



C O M M O N. 



Common 

 Measure?. 



+ 12/" — 2 X V») in an octave, and that he had " thence 

 deduced an ingenious invention of finding and applying 

 a least common measure to all harmonic intervals, not 

 precisely perfect, but very near it." 



Of the method or process by which Mercator obtain- 

 ed his series of artificial commas, answering to the 1 9 

 intervals, which Dr Holder has given as an extract from 

 the manuscript, we are quite uninformed ; but to us it 

 seems probable, from the relation, that having calcula- 

 ted the number of commas to the nearest whole number 

 in each case, answering to the 19 intervals mentioned, 

 and found them as follows, (see Plate XXX. Vol. II.) 

 viz. c—\, E=2, cf=3, 8=4, S=5, S=6, 1=8, T=9; 

 3rd=15, 111=18, 4th=23, IV=27, and5th=28; V= 

 33, 6th=38, VI=41, and 7th=46; and VII=51,and 

 VIII =56 ; and trying the relations of these by addi- 

 tion and subtraction of these commas, he thus found, 

 that all those from c to T inclusive, exactly answered 

 to the known and corresponding values and relations 

 of these intervals, as found by the multiplication of the 

 terms of their ratios, according to the equations given 

 in our articles Apotome, Comma, &c. ; and that again, 

 from the 3rd to the 5th inclusive, the relations of these 

 commas were proper to each other, though each was an 

 unit too much, when compared by the proper addi- 

 tions and subtractions with the first part of the series ; 

 and, further, that from the V th to the 7 th inclusive, the 

 relations were true with respect to each other, but 

 each was 1 comma more than the last part of the se- 

 ries, and 2 commas more than calculations from the 

 first part would have given them ; and, lastly, that the 

 VII th and VIII th were 3 commas too great for compa- 

 rison with the first part of the series : and thus proba- 

 bly it was, that the ingenious Mercator was led to re- 

 duce all the numbers to the first part, and so deduced 

 his series of artificial commas, i , 2, 3, 4, 5, 6, 8, and 

 9; 14, 17, 22, 26, and 28; 31, 36, 39, and 44; 48 and 

 53 ; answering to the 1 9 intervals above mentioned, as 

 Or Holder has given them. 



We are not aware, that this mode of accounting for 

 Mercator's unexpected discovery of the curious proper- 

 ties of the above series, was ever before published, or 

 that the same attracted the notice of any curious per- 

 son in these inquiries after Dr Holder, until the year 

 1807, when Mr John Farey sen. having extracted all 

 the intervals contained in the Overend, and other ma- 

 nuscripts which he had perused, and, for the conve- 

 nience of future reference, had, with considerable la- 

 bour, reduced them all into one notation, by the inter- 

 vals marked 2, f, and m, as shewn in part in Plate V. 

 in vol. xxviii. of the Philosophical Magazine, and in 

 our 30th Plate in Vol. II. He then quickly discover- 

 ed, that the number of m's in the last column of his 

 Table, answered exactly to Mercator's number affixed 

 to the same interval in Dr Holder's Treatise ; and not 

 only shewed the reason thereof, but his table of inter- 

 vals, so arranged, furnished a far more accurate and ex- 

 tensive set of artificial commas, or under commas 612, 

 to the octave, by extracting the number of 2's in the 

 first column. 



It has also been shewn by that gentleman, that each 

 of the columns of any notation of small terms, or inter- 

 vals, so that negative signs are avoided, except when 

 they affect a whole column, furnishes a set of artificial 

 commas, as must, indeed, be evident, since there can 

 be no carrying forwards of whole numbers from one 

 column of notation to another, in adding, or borrow- 

 ing in performing subtraction, as in the columns of 

 pounds shillings and pence in money calculations, &c; 



but each column, when there are no decimals, concurs Common 

 in shewing, independently, the same results, as far as Meas ures- 

 they can be expressed therein ; yet carrying and bor- — "r-=— 

 rowing are sometimes used, at the rate of .14966096, 

 &c. between/ and 2, and of .007862405, &c. between 

 m and 2. 



The m and f, in Mr Farey's notation, are the small- 

 est intervals that are yet known, we believe ; but two 

 others, d and F, occur in our Table, Plate XXX. Vol. II. 

 between the latter and 2, the largest term of this no- 

 tation; which term was fixed upon, on account of the 

 many important relations which the schisma (2) bears 

 to other intervals ; but Mr Farey has tried other nota- 

 tions of these small intervals, as follows, viz. 61 2</+ 

 1848/"— 559w=VIII, 6I2F + 624/— 559'»=VIII, 306r 

 — 294/+ 53m= VIII, 5592 + 5:^+17)/= VIII, 279^r 

 + 53-V— 1084/= VIII, 65r + 4292 + 53F= VIII, 65 c 

 _12R— f.]2=VIII, 55^c+ 12/"-2-V™=VIII, 53c+ 

 292 + 12/=VIII=53c+12r + 52=VIII, 12<p + 41c + 

 412=VIII, i2/c + 53c— 312=VIII, 12jfcx53c— 192 

 =VIII, 12^ + 41c— 192=VIII, 12D = 41c— 72= 

 VIII, 12/ + 29€+222=VIII, &c. Whence seve- 

 ral different sets of artificial commas, of 1848, 624, 

 612, 559, 429, 306, 294, 279i 171, 108^ 65, 55 T \, 

 53, 41, 31, 29, 22, 19, 12, &c. respectively, in the 

 octave, might now readily be calculated by that gen- 

 tleman's manuscript tables of intervals. The first of 

 which, 1848, would give a set of artificial commas, 

 considerably more exact in the very smaller intervals, 

 owing to the largeness of the numbers, than that by 

 2, adopted by Mr Farey; but the notation, whence 

 it is derived, for other reasons besides the negative 

 sign in all its terms, is less adapted to general use 

 than 2, f, and m, and they would not prove in the 

 least degree more exact, for all intervals larger than 

 2. The notation that is adopted, besides furnish- 

 ing Mercator's and Farey's artificial commas, contains 

 another set of these (or of artificial half notes rather) 

 in its middle column, which shews the number of de- 

 grees, or half-notes, or the finger-key, (J 2 in the oc- 

 tave,) to which any interval belongs, and which is of 

 very considerable use in the practice of musical com- 

 putations. 



The musical student must, however, be on his guard 

 in using the above, and aU other artificial commas, in 

 his calculations ; always remembering, that when whole 

 numbers of one interval only are used, that the same 

 have no fixed or precise values, but vary, within small 

 Umits, (if the number in the octave is considerable,) in 

 expressing every different interval to which they are 

 applied. Thus, if 1 of Farey's artificial commas be 

 supposed exactly to represent the schisma, or 2, (and 

 these commas cannot represent any smaller interval) : 

 then in 2, the alto comma or 1 is 1.000000 2; but in 

 10 for the minor comma, or C, the same is 1.00078n24 ; 

 in 11 for c, each is 1.0007020; in 12 for A, each is 

 1,0006552 ; in 21 for £, each is 1.007488; in 57 for S, 

 each is 1.0033154; in 104 for T, each is 1.0035585; 

 in 197 for III, each is 1.0037173; in 358 for V, each 

 is 1.0036072; in 6)2 for VIII, each is 1.0036154, &c. 

 differing in every interval, and so they would do, on 

 the supposition that ^ C, T 'j c, T ' T d", &c. any one of 

 them exactly represented this artificial comma ; and yet 

 the curious and admirable properties of these numbers 

 are such, that, by adding and subtracting, they truly re- 

 present, and with the utmost facility perform, all the 

 operations with intervals, as certainly as the logarithms, 

 or the multiplication or division of the ratios can do, 

 except with intervals smaller than, or very nearly equal 



